Proving that the sequence a 1 = 4, a n + 1 = a n + 20 converges and finding its limit

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Proving that the sequence       a    1    =  4,       a          n      +      1        =            a              n              +    20   converges and finding its limit

uavklarajo · Accepted Answer

First let&#039;s guess what the limit is (if exists): say       a    n    →  α, then, by continuity of the operations + and               , we have  α  =      α    +    20    ,      i.e.,          α    2    =  α  +  20Its roots are -4 and 5. Since       a    n    &amp;gt;  0 always, only   α  =  5 can be valid.Now use induction, with additional hypothesis that       a    n    ≤  5 to prove       a          n      +      1        &amp;gt;      a    n   and still       a          n      +      1        ≤  5

Proving that the sequence a_1=4, a_(n+1)=sqrt(a_n+20) onverges and finding its limit

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